For ex, I’m trying to intuitively understand why the derivative of secx = secxtanx as opposed to just memorizing it.
When it comes to the derivatives of trig functions, are we still looking at the slope of the tangent line of f(x)? Not sure if this question makes sense. Thanks!
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I’ve put up [a little Desmos graph to demonstrate](https://www.desmos.com/calculator/3ttc6n3bhk). On that graph, you can see, the tangent and secant together with a unit circle. You can see, that “tangent” is really a tangent to circle, and “secant” is where the tangent inter**sects** the X axis.
The question of derivative basically boils down to the question: if we make a little nudge `dx` to the `x`, how much does `sec x` grows? In other words, how much is `d(sec x)`? You can control `dx` on the graph with the first slider (labeled `ϕ_d`). You can also use the “play” button on the left of it to play an animation.
The change to `sec x` is comprised of 3 factors:
* Rotation of tangent line
* “Extension” of tangent line, until it intersects the secant.
* Movement of point P around the circle
Let’s consider them one by one:
**Step 0.** Movement of point P around the circle can be ignored. For small `dx`, its contribution to the result is negligible. We can just pretend, that tangent line just rotates in place.
**Step 1.** Let’s consider, that tangent line rotates, but doesn’t extend. Where the secant point S0 will be? Well, it rotates `dx` radians around P, on a circle of radius `tan x`. So, it moves away `tan x*sin dx` distance. Because `dx` is small, `sin dx ~= dx`, so we get `tan x * dx`. Notice, that movement is perpendicular to original tangent line. That means it parallel to radius OP (because OP is also parallel to tangent).
*Step 1 conclusion.* S0 moves distance `tan x * dx` parallel to OP.
**Step 2**. Now let’s consider, that tangent moves with S0 and extends, but doesn’t rotate. That means, we “move” tangent line `tan x * dx` away from the circle. Radius OP = 1, but now it grows to `1 + tan x * dx`, so it scales by a factor of `1 + tan x * dx`. That means, that secant OS should also grow by a factor `1 + tan x * dx` (from triangle POS, it’s angles don’t change, so it stays similar to itself). So `sec x + d(sec x) = sec x * (1 + tan x * dx) = sec x + sec x * tan x * dx`, which means `d(sec x) = sec x * tan x * dx`.
So, derivative of `sec x` is `d(sec x)/dx = sec x * tan x`. The `tan x` is from rotating the tangent line without extending, `sec x` is from moving and extending tangent line without rotating.
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